A SIMPLE DEFINITION OF THE FEYNMAN INTEGRAL, WITH APPLICATIONS 11
OBSERVATION: Since each functional F in S or S or S determines a unique
measure ief?l and each measure * e ft determines a unique equivalence class of function
als in each of the three spaces S,S , and S , the three Banach algebras are pairwise
isomorphic.
REMARK U:Let veBV[a,b] and let x be absolutely continuous on [a,b]. Then the
following RiemannStieltjes and Lebesgue integrals are equal:
J v(t)dx(t) = J v(t) &&• dt.
a a
This follows from the definition of the RiemannStieltjes integral and the fact that
var x is a uniformly continuous function of t on [a,b] .
[a,t]
LEMMA 2.6. S' 5 S * Q S n s .
PROOF:For F e s 1 , there exist s n* €J^» (definitions given in [k]) so tha t
v b
F(x) = P expCi E J v (t)dx (t)]dH'(v)
(BV)V
^
= l a
"
3
for x€ C [a,b] . Just as in the proof of Theorem 3.0 in [k], we define a measure H on
Lg[a,b] as follows. Let E e B(L^) , then set "(E) SH« (E fl
(BV)V)
. Let x e D [ a,b ] ,
then by Remark 2 of [6] and Remark h we have for v e BV ,
J v(t)dx(t ) = J" v(t ) H dt = J v(t)dx(t ) •
a a a
Thus for x€ D ,

v b
F(x) = J exp£i Z J1 v.(t)dx.(t))dH'(v )
(BV)V
^ *
J 3
v b dx.
= J exp!i E J v (t) jJ dt}d*'(v)
(BV)V
* =
l a
v b dx. _
= J
v
exp{i S J v (t) ^ d t } d H ( v )
L2 j = l a
J
v b
= J1 expfi S J v.(t)dx.(t)}dK(v) .
J L^ 0=1 a J J
By Theorem 3.0 of [k]9 the f i r s t and l a s t members above are equal for salmost a l l
v
r
* *
x e C [a,b] , so tha t S'crs \ S =s by definition .
We now present an example which shows that S / S H S .